metacyclic, supersoluble, monomial, 2-hyperelementary
Aliases: C84.1C4, C4.Dic21, C28.50D6, C4.15D42, C21⋊7M4(2), C12.51D14, C28.1Dic3, C12.1Dic7, C22.Dic21, C84.57C22, C21⋊C8⋊5C2, (C2×C42).3C4, (C2×C28).5S3, (C2×C84).7C2, (C2×C12).5D7, (C2×C4).2D21, C42.30(C2×C4), C3⋊2(C4.Dic7), C7⋊2(C4.Dic3), (C2×C6).3Dic7, C6.7(C2×Dic7), C14.7(C2×Dic3), C2.3(C2×Dic21), (C2×C14).3Dic3, SmallGroup(336,96)
Series: Derived ►Chief ►Lower central ►Upper central
Generators and relations for C84.C4
G = < a,b | a84=1, b4=a42, bab-1=a-1 >
Smallest permutation representation of C84.C4
►On 168 points
Generators in S168
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84)(85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168)
(1 94 22 157 43 136 64 115)(2 93 23 156 44 135 65 114)(3 92 24 155 45 134 66 113)(4 91 25 154 46 133 67 112)(5 90 26 153 47 132 68 111)(6 89 27 152 48 131 69 110)(7 88 28 151 49 130 70 109)(8 87 29 150 50 129 71 108)(9 86 30 149 51 128 72 107)(10 85 31 148 52 127 73 106)(11 168 32 147 53 126 74 105)(12 167 33 146 54 125 75 104)(13 166 34 145 55 124 76 103)(14 165 35 144 56 123 77 102)(15 164 36 143 57 122 78 101)(16 163 37 142 58 121 79 100)(17 162 38 141 59 120 80 99)(18 161 39 140 60 119 81 98)(19 160 40 139 61 118 82 97)(20 159 41 138 62 117 83 96)(21 158 42 137 63 116 84 95)
G:=sub<Sym(168)| (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,94,22,157,43,136,64,115)(2,93,23,156,44,135,65,114)(3,92,24,155,45,134,66,113)(4,91,25,154,46,133,67,112)(5,90,26,153,47,132,68,111)(6,89,27,152,48,131,69,110)(7,88,28,151,49,130,70,109)(8,87,29,150,50,129,71,108)(9,86,30,149,51,128,72,107)(10,85,31,148,52,127,73,106)(11,168,32,147,53,126,74,105)(12,167,33,146,54,125,75,104)(13,166,34,145,55,124,76,103)(14,165,35,144,56,123,77,102)(15,164,36,143,57,122,78,101)(16,163,37,142,58,121,79,100)(17,162,38,141,59,120,80,99)(18,161,39,140,60,119,81,98)(19,160,40,139,61,118,82,97)(20,159,41,138,62,117,83,96)(21,158,42,137,63,116,84,95)>;
G:=Group( (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84)(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168), (1,94,22,157,43,136,64,115)(2,93,23,156,44,135,65,114)(3,92,24,155,45,134,66,113)(4,91,25,154,46,133,67,112)(5,90,26,153,47,132,68,111)(6,89,27,152,48,131,69,110)(7,88,28,151,49,130,70,109)(8,87,29,150,50,129,71,108)(9,86,30,149,51,128,72,107)(10,85,31,148,52,127,73,106)(11,168,32,147,53,126,74,105)(12,167,33,146,54,125,75,104)(13,166,34,145,55,124,76,103)(14,165,35,144,56,123,77,102)(15,164,36,143,57,122,78,101)(16,163,37,142,58,121,79,100)(17,162,38,141,59,120,80,99)(18,161,39,140,60,119,81,98)(19,160,40,139,61,118,82,97)(20,159,41,138,62,117,83,96)(21,158,42,137,63,116,84,95) );
G=PermutationGroup([[(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84),(85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168)], [(1,94,22,157,43,136,64,115),(2,93,23,156,44,135,65,114),(3,92,24,155,45,134,66,113),(4,91,25,154,46,133,67,112),(5,90,26,153,47,132,68,111),(6,89,27,152,48,131,69,110),(7,88,28,151,49,130,70,109),(8,87,29,150,50,129,71,108),(9,86,30,149,51,128,72,107),(10,85,31,148,52,127,73,106),(11,168,32,147,53,126,74,105),(12,167,33,146,54,125,75,104),(13,166,34,145,55,124,76,103),(14,165,35,144,56,123,77,102),(15,164,36,143,57,122,78,101),(16,163,37,142,58,121,79,100),(17,162,38,141,59,120,80,99),(18,161,39,140,60,119,81,98),(19,160,40,139,61,118,82,97),(20,159,41,138,62,117,83,96),(21,158,42,137,63,116,84,95)]])
90 conjugacy classes
| class | 1 | 2A | 2B | 3 | 4A | 4B | 4C | 6A | 6B | 6C | 7A | 7B | 7C | 8A | 8B | 8C | 8D | 12A | 12B | 12C | 12D | 14A | ··· | 14I | 21A | ··· | 21F | 28A | ··· | 28L | 42A | ··· | 42R | 84A | ··· | 84X |
| order | 1 | 2 | 2 | 3 | 4 | 4 | 4 | 6 | 6 | 6 | 7 | 7 | 7 | 8 | 8 | 8 | 8 | 12 | 12 | 12 | 12 | 14 | ··· | 14 | 21 | ··· | 21 | 28 | ··· | 28 | 42 | ··· | 42 | 84 | ··· | 84 |
| size | 1 | 1 | 2 | 2 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 42 | 42 | 42 | 42 | 2 | 2 | 2 | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 | 2 | ··· | 2 |
90 irreducible representations
| dim | 1 | 1 | 1 | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 | 2 |
| type | + | + | + | + | - | + | - | + | - | + | - | + | - | + | - | ||||||
| image | C1 | C2 | C2 | C4 | C4 | S3 | Dic3 | D6 | Dic3 | D7 | M4(2) | Dic7 | D14 | Dic7 | D21 | C4.Dic3 | Dic21 | D42 | Dic21 | C4.Dic7 | C84.C4 |
| kernel | C84.C4 | C21⋊C8 | C2×C84 | C84 | C2×C42 | C2×C28 | C28 | C28 | C2×C14 | C2×C12 | C21 | C12 | C12 | C2×C6 | C2×C4 | C7 | C4 | C4 | C22 | C3 | C1 |
| # reps | 1 | 2 | 1 | 2 | 2 | 1 | 1 | 1 | 1 | 3 | 2 | 3 | 3 | 3 | 6 | 4 | 6 | 6 | 6 | 12 | 24 |
Matrix representation of C84.C4 ►in GL4(𝔽337) generated by
| 236 | 321 | 0 | 0 |
| 28 | 248 | 0 | 0 |
| 0 | 0 | 220 | 98 |
| 0 | 0 | 0 | 72 |
| 162 | 17 | 0 | 0 |
| 161 | 175 | 0 | 0 |
| 0 | 0 | 314 | 31 |
| 0 | 0 | 76 | 23 |
G:=sub<GL(4,GF(337))| [236,28,0,0,321,248,0,0,0,0,220,0,0,0,98,72],[162,161,0,0,17,175,0,0,0,0,314,76,0,0,31,23] >;
C84.C4 in GAP, Magma, Sage, TeX
C_{84}.C_4 % in TeX
G:=Group("C84.C4"); // GroupNames label
G:=SmallGroup(336,96);
// by ID
G=gap.SmallGroup(336,96);
# by ID
G:=PCGroup([6,-2,-2,-2,-2,-3,-7,24,121,50,964,10373]);
// Polycyclic
G:=Group<a,b|a^84=1,b^4=a^42,b*a*b^-1=a^-1>;
// generators/relations
Export